The path algebra of the D₄ quiver

The D₄ quiver has seven paths: the four trivial ones and the three arrows. So its path algebra has rank seven over any semiring satisfying the strong rank condition. Over a division ring it is seven-dimensional. Finite-dimensionality needs nothing specific to this quiver: it is acyclic, so TauCeti.finiteDimensional_pathAlgebra_of_isAcyclic applies via TauCeti.Quiver.D4.isAcyclic.

Main results

• TauCeti.Quiver.D4.card_totalPath and TauCeti.Quiver.D4.finrank_pathAlgebra: there are seven paths, so the path algebra has rank seven.

References

See Derksen--Weyman, An Introduction to Quiver Representations, and Assem--Simson--Skowroński, Elements of the Representation Theory of Associative Algebras I, Ch. II.

theorem TauCeti.Quiver.D4.card_totalPath : Fintype.card (TotalPath D4) = 7

The D₄ quiver has seven paths: the four trivial ones and the three arrows.

@[simp]

theorem TauCeti.Quiver.D4.finrank_pathAlgebra (k : Type w) [Semiring k] [StrongRankCondition k] : Module.finrank k (pathAlgebra k D4) = 7

The path algebra of the D₄ quiver has rank seven: four trivial paths and three arrows.